ON LOCAL TORUS ACTIONS MODELED ON THE STANDARD REPRESENTATION(The theory of transformation groups and its applications)

ON LOCAL

TORUS

ACTIONS

MODELED

ON THE STANDARD

REPRESENTATION

TAKAHIKO YOSHIDA

1. INTRODUCTION

Let

$S^{1}$ be the unit circle $\bm{t}dT^{n}$ $:=(S^{1})^{n}$ the $n$-dimensional compact

torus.

$T^{n}$ acts

on

the_$n$-dImensionalcomplex vectorspace$\mathbb{C}^{n}$ by coordinate-wisecomplex

multiplication. This action is called the standard repoesentation$ofT^{n}$

.

$T^{n}$ acts

on a

complex$n$-dimensionaltoric variety$X$

as

asubgroup of$(\mathbb{C}^{n})^{*}$

.

If$X$ is _{$non\sin_{1^{1ar}}$}, then for each point $x\in X$, there exists acoordinate neighborhood $(U, \rho, \varphi)$ of $x$,

where where $U$ is

a

$T^{n}$-invariant openset of_$X,$

$\rho$ is

an

automorphism of$T^{n}$, and $\varphi$ is

a

$\rho$-equivariant diffeomorphism from $U$ to

some

$T^{n}$-invariant open subset in $\mathbb{C}^{n}$. In general, a $T^{n}$-action

on

a2_$n$-dimensional manifold which is covered by

such coordInate $neighb_{01}\cdot hoods$ is said to be locally standard. See $[4, 2]$ _for

more

details. This property is

one

of the starting point of their pioneer work [4] of

Davis-Januszkiewicz and now, it playsa central role in toric topology.

Asimilar structure can be

seen

in Lagrangian fibrations. Let $(X,\omega)$ be

a2n-dimensional smooth symplecticmanifold $\bm{t}dB$

an

$n$-dimensionalsmooth manifold

with

corners.

We call amap $\mu:(X, \omega)arrow B$ alocally $tor\dot{b}C$ Lagmngian

fibmtion

if $\mu$ is locally identified with the moment map of the standard representation of

$T^{n}$

.

It is known that there exists

an

atlas $\{(U_{\alpha}, \varphi_{\alpha})\}$ of$X$ and there also exists an

automorphism $\rho$ of$T^{n}$

on

each nonemptyoverlap $U_{\alpha}\cap U_{\beta}$ such that each

$\varphi_{\alpha}$ sends

$U_{\alpha}diff\infty morphically$to

some

$T^{n}$-invariant open subset of$\mathbb{C}^{n}$ and the overlapmap

$\varphi_{\alpha}^{X}o(\varphi_{\beta}^{X})^{-1}$ is $\gamma equivariant$ (see also Example 2.9).

In [13], as ageneralizationof alocally standard torus action and also as an

un-derlyingstructure of alocallytoric Lagrangian fibration,

we

introduced the notion

of alocal$T^{n}$-action modeledonthe standard representation, anddefinedtwo

topo-logical invariants calledthecharacteristic pair_{and the Euler class of the orbit map}

for alocal $T^{n}$-action, thenproved that local $T^{n}$-actions

are

topologically classified

by these two invariants. We also investigate the symplectic

case.

The content of [13] is arefinement of the work [12].

This isan announcement of [13]. In thenext section, werecallthe definition and basic facts of alocal$T^{n}$-action. In Section 3,

we

explain that alocal $T^{n}$-action is

accompaniedbythe principal$Aut(T^{n})$-bundle and thecharacterlstic bundle. $Af\mathfrak{b}er$

that,

we

recall the construction of the canonical model of alocal $T^{n}$-action. In

Section 4,

we

define the Euler class of the orbit map. Section 5is devoted to the

topological classification of local $T^{n}$-actions. Theorem 5.1 is the main theorem

of the first part of this paper. We also describe the idea of the proof, where the

canonical model plays

an

importantrole. As acorollary, we

can

obtain that locally

standard $T^{n}$-actions

are

claesified by the characteristic bundle and the Euler class

of the orbit map up to equivariant homeomorphisms (Corollary 5.2). One ofthe

important examples of manifolds equipped with local $T^{n}$-actions is alocally torIc

Lagrangian fibration with $n$-dimensional fibers. Finally, in Section 6,

we

give the

2000 Mathematics Subject _{Classification.} Primary $57R15_{j}Se\omega ndary57S99,55R65$

.

Keywords andphrases. Local torus actions, locallystandardtorus actions.

necessaryand sufficientcondition that amanifold withalocal$T^{n}$-action becom

es

a

locally toric Lagrangian fibration and also describethe classification of$10$callytoric

Lagrangian fibrations.

Throughout this paper we employ thevector notation in order to represent

ele-ments of$\mathbb{C}^{n}$, namely, _{$z=(z_{1}, \ldots, z_{n})\in \mathbb{C}^{n}$}

.

The similar notation is also used for

$T^{n}=(S^{1})^{n},$ $\mathbb{R}^{n}$, etc.

2.

DEFINITIONS

AND BASIC FACTS

Let $X$ be

a

paracompact, Hausdorffspace.

Definition 2.1. A weakly standard $C^{r}(0\leq r\leq\infty)$ atlas of $X$ is

an

atlas

$\{(U_{\alpha}^{X}, \varphi_{\alpha}^{X})\}_{\alpha\in A}$ which satisfies the following properties

(1) foreach$\alpha,$$\varphi_{\alpha}^{X}$ is

a

homeomorphismfrom $U_{\alpha}^{X}$ to

an

openset of C’ invariant

under the standard representation of$T^{n}$ and

(2) _{for each nonempty overlap} $U_{\alpha\beta}^{X}$ $:=U_{\alpha}^{X}\cap U_{\beta}^{X}$, there exists

an

automorphism

$\rho_{\alpha\beta}$ of$T^{n}$

as

a Lie group such that the overlap map $\varphi_{\alpha\beta}^{X}$ $:=\varphi_{\alpha}^{X}\circ(\varphi_{\beta}^{X})^{-1}$

is $\rho_{\alpha\beta^{-}}equivariantC$‘ diffeomorphic with respect to the restrictions ofthe standardrepresentationof$T^{n}$ to$\varphi_{\alpha}^{X}(U_{\alpha\beta}^{X})$and$\varphi_{\beta}^{X}(U_{\alpha\beta}^{X})$

.

(Thelatter

means

that $\varphi_{\alpha\beta}^{X}(u\cdot z)=\rho_{\alpha\beta}(u)\cdot\varphi_{\alpha\beta}^{X}(z)$ for$u\in T^{n}$ and $z\in\varphi_{\beta}^{X}(U_{\alpha\beta}^{X}).)$

Two weakly standard $C^{r}$ atlases $\{(U_{\alpha}^{X}, \varphi_{\alpha}^{X})\}_{\alpha\in A}$ and $\{(V_{\beta}^{X},\psi_{\beta}^{X})\}_{\beta\in B}$ of$X^{2n}$

are

equivalentifoneachnonempty overlap $U_{\alpha}^{X}\cap V_{\beta}^{X}$, there exists an automorphism$\rho$of

$T^{n}$ such that$\varphi_{\alpha}^{X}\circ(\psi_{\beta}^{X})^{-1}$is$\rho$-equivariant$C^{r}$ diffeomorphic. Wecall

an

equivalence

class of weakly standard $C^{r}$ atlases a $C^{r}$ local $T^{n}$-action on $X^{2n}$ modeled on the

standard representation and denote it by $\mathcal{T}$

.

In the rest of this paper, a $C^{r}$ local $T^{n}$-action

on

$X^{2n}$ modeled

on

the standard

representation is$of\mathfrak{b}en$ called

a

$C^{r}$ local $T^{n}$-actionon $X^{2n}$, or

more

$s$imply, a $1o$cal

$T^{n}$-action

on

$X$ if there are no confusions.

Let (X,$\mathcal{T}$) bea$2n$-dimensional manifold_$X$equippedwith

a

$C^{r}$local$T^{n}$-action$\mathcal{T}$

and $\{(U_{\alpha}^{X}, \varphi_{\alpha}^{X})\}_{\alpha\in A}$ amaximalweakly standard atlas of_$X$ which belongs to$\mathcal{T}$. For

(X,$\mathcal{T}$)

we

cangeneralizethe orbit space and the orbit mapin

the following way. We endow each quotient space $\varphi_{\alpha}^{X}(U_{\alpha}^{X})/T^{n}$ with the quotient topology induced $hom$ the topology of $\varphi_{\alpha}^{X}(U_{\alpha}^{X})$ by the natural projection $\pi:\varphi_{\alpha}^{X}(U_{\alpha}^{X})arrow\varphi_{\alpha}^{X}(U_{\alpha}^{X})/T^{n}$

.

By the property (2) for each overlap $U_{\alpha\beta}^{X},$ $\varphi_{\alpha\beta}^{X}$ induces ahomeomorphism $hom$

$\varphi_{\beta}^{X}(U_{\alpha\beta}^{X})/T^{n}$ to $\varphi_{\alpha}^{X}(U_{\alpha\beta}^{X})/T^{n}$

.

We define two elements $b_{\alpha}\in\varphi_{\alpha}^{X}(U_{\alpha}^{X})/T^{n}$ and $b_{\beta}\in\varphi_{\beta}^{X}(U_{\beta}^{X})/T^{n}$

are

equivalent if$b_{\alpha}\in\varphi_{\alpha}^{X}(U_{\alpha\beta}^{X})/T^{n},$ $b_{\beta}\in\varphi_{\beta}^{X}(U_{\alpha\beta}^{X})/T^{n}$ and the map induced by $\varphi_{\alpha\beta}^{X}$ sends $b_{\beta}$ to $b_{\alpha}$. It is an equivalence relation

on

the disjoint

union $II_{\alpha}(\varphi_{\alpha}^{X}(U_{\alpha}^{X})/T^{n})$

.

We call the quotient space of$LI_{\alpha}(\varphi_{\alpha}^{X}(U_{\alpha}^{X})/T^{n})$ bythe

equivalence relation together with aquotient topology the orbit space of the local

$T^{n}$-action $\mathcal{T}$

on

_$X$ and denote it by

$B_{X}$

.

It iseasy to

see

that $Bx$ is aHausdorff

space and $\{\varphi_{\alpha}^{X}(U_{\alpha}^{X})/T^{n}\}_{\alpha\in A}$ is

an

open covering of $Bx$

.

By the construction of $B_{X}$, the map $LI_{\alpha}\pi 0\varphi_{\alpha}^{X}$: $U_{\alpha}^{U_{\alpha}^{X}}arrow II_{\alpha}(\varphi_{\alpha}^{X}(U_{\alpha}^{X})/T^{n})$ induces the map from $X$

to $B_{X}$

.

We call it the orbit mapof the local $T^{n}$-action $\mathcal{T}$

on

$X$ and denote it by

$\mu x:Xarrow B_{X}$

.

Notice that bythe construction, it is acontinuous open map. Let $\mathbb{R}_{+}^{n}$ be the standard$n$-dimensional positive

cone

$\mathbb{R}_{+}^{n}:=\{\xi=(\xi_{1}, \ldots, \xi_{n})\in \mathbb{R}^{n} : \xi_{i}\geq 0i=1, \ldots,n\}$

.

It has the natural stratificationwithrespect to the number of coordinates $\xi_{i}$ which

are

equalto

zero.

Deflnition 2.2. Let $B$ be a Hausdorff space. A structure

of

an

n-dimensional

ontoopen subsetsof$\mathbb{R}_{+}^{n}$

so

that overlap

maps are

homeomorphismswhich preserve

the natural stratifications induced from the

one

of $\mathbb{R}_{+}^{n}$. See [3, Section 6] for

a

topological manifold with corners.

Proposition 2.3. $B_{X}$ is endowed with astructure

of

an n-dimensional topological

manifold

with

comers.

Proof.

We definethe map $\mu_{\mathbb{C}^{\mathfrak{n}}}$: $\mathbb{C}^{n}arrow \mathbb{R}^{n}$ by

(2.1) $\mu_{C^{n}}(z)=(|z_{1}|^{2}, \ldots, |z_{n}|^{2})$

for $z=$ $(z_{1}, \ldots , z_{n})\in \mathbb{C}^{n}$

.

Notice

that the image of

$\mu_{\mathbb{C}^{n}}$ is the n-dimensional

standard positive

cone

$\mathbb{R}_{+}^{n}$

.

It is invariant under the standard representationof$T^{n}$

and induces the homeomorphism from $\mathbb{C}^{n}/T^{n}$ to $\mathbb{R}_{+}^{n}$

.

The orbit space _{$\mathbb{C}^{n}/T^{n}$} is

endowedwith the natural stratification whose k-dimensional stratum consists of

k-dimensional orbitsand the homeomorphism induced by$\mu_{C^{n}}$ preserves stratifications

of$\mathbb{C}^{n}/T^{n}$ and$\mathbb{R}_{+}^{n}$

.

We put $U_{\alpha}^{B}$ $:=\varphi_{\alpha}^{X}(U_{\alpha}^{X})/T^{n}$

.

The restriction of

$\mu_{\mathbb{C}^{n}}$ to $\varphi_{\alpha}^{X}(U_{\alpha}^{X})$

induces the homeomorphism from $U_{\alpha}^{B}$ to the open subset $\mu_{C^{n}}(\varphi_{\alpha}^{X}(U_{\alpha}^{X}))$ of

$\mathbb{R}_{+}^{n}$,

which isdenotedby$\varphi_{\alpha}^{B}$

.

By theconstruction,

on

each overlap

$U_{\alpha\beta}^{B}$ $:=U_{\alpha}^{B}\cap U_{\beta}^{B}$, the overlap map $\varphi_{\alpha\beta}^{B}$ $:=\varphi_{\alpha}^{B}o(\varphi_{\beta}^{B})^{-1}$: $\mu_{C^{n}}(\varphi_{\beta}^{X}(U_{\alpha\beta}^{X}))arrow\mu_{C^{n}}(\varphi_{\alpha}^{X}(U_{\alpha\beta}^{X}))$ preserves the

natural stratifications of$\mu_{\mathbb{C}^{n}}(\varphi_{\alpha}^{X}(U_{\alpha\beta}^{X}))$and $\mu_{\mathbb{C}^{n}}(\varphi_{\beta}^{X}(U_{\alpha\beta}^{X}))$

.

Thus,

$\{(U_{\alpha}^{B}, \varphi_{\alpha}^{B})\}_{\alpha\in A,\square }$ is the desired atlas.

Remark 2.4. The atlas $\{(U_{\alpha}^{B}, \varphi_{\alpha}^{B})\}_{\alpha\in A}$of$B_{X}$ construct$ed$ in the proofof

Proposi-tion 2.3 has following properties

(1) for $e$ach $\alpha,$ $U_{\alpha}^{X}=\mu_{X}^{-1}(U_{\alpha}^{B}),$ $\varphi_{\alpha}^{X}(U_{\alpha}^{X})=\mu_{\mathbb{C}^{n}}^{1}(\varphi_{\alpha}^{B}(U_{\alpha}^{B}))$ and the following diagram commutes $BX\downarrow$$\mu x$ $\supset\mu_{X}^{-1}|$ $x$ $\supset$ _$U$ $U_{\alpha}^{B})arrow\mu_{C^{n}}^{1}(\varphi_{\alpha}^{B}\varphi_{\alpha}^{X}$ $\alpha B_{arrow\varphi_{\alpha}^{B}(}^{\varphi_{\alpha}^{B}}\mu_{X}\downarrow$ $\mu c(U_{\alpha}^{B}))\subset n$ $\mathbb{R}\mathbb{C}\downarrow$ $U_{\alpha}^{B})$ $\subset$ $n$ $\mu_{C}n$ $n+$

(2) the restriction of $\{(U_{\alpha}^{B}, \varphi_{\alpha}^{B})\}_{\alpha\in A}$to the interior $Bx\backslash \partial B_{X}$ of $B_{X}$ is a $C^{r}$

atlas of$B_{X}\backslash \partial B_{X}$

.

Let $(X_{1},\mathcal{T}_{1})$ and $(X_{2},\mathcal{T}_{2})$ be$2n$-dimensional manifolds$X_{1}$ and$X_{2}$ equipped with

$C^{r}$ local$T^{n}$-actions$\mathcal{T}_{1}$ and $\mathcal{T}_{2}$

.

Let $\{(U_{\alpha}^{X_{1}}, \varphi_{\alpha}^{X_{1}})\}_{\alpha\in A}$ and $\{(U_{\beta}^{X_{2}}, \varphi_{\beta}^{X_{2}})\}_{\beta\in \mathcal{B}}$ bethe maximal weakly standard atlases of$X_{1}$ and $X_{2}$ which belongto $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$

.

Deflnition 2.5. $(X_{1}, \mathcal{T}_{1})$ and $(X_{2},\mathcal{T}_{2})$ are $C^{r}$ isomorphicif there exists

a

$C^{r}$ dif-feomorphism $fx:X_{1}arrow X_{2}$ from $X_{1}$ to $X_{2}$ and on each nonempty overlap $U_{\alpha}^{X_{1}}\cap$

$(f_{X})^{-1}(U_{\beta}^{X_{2}})\neq\emptyset$there exists

an

automorphism$\rho$of$T^{n}$such that$\varphi_{\beta}^{X_{2}}\circ f_{X}\circ(\varphi_{\alpha}^{X_{1}})^{-1}$

is $\rho$-equivariant. We also call such a $C^{r}$ diffeomorphism $f_{X}$ a $C^{r}$ isomorphismand

denote it by $f_{X}$: $(X_{1},\mathcal{T}_{1})arrow(X_{2},\mathcal{T}_{2})$

.

Notice that a$C^{r}$ isomorphism $fx:(X_{1}, \mathcal{T}_{1})arrow(X_{2}, \mathcal{T}_{2})$induces the stratification

preserving $hom\infty morphismf_{B}$: $Bx_{1}arrow Bx_{2}$ between their orbit spaces such that

$f_{X}$ and $f_{B}$ satisfy $\mu_{X_{2}}\circ f_{X}=f_{B}\circ\mu x_{1}$.

We give examples of local torus actions.

Example 2.6 (Locally

standard

torus actions). Let $T^{n}$ act smoothly

on a

2$r\vdash$

dimensional smoothmanifold $X$

.

A standard coordinate neighborhoodof$X$ consists

ofatriple $(U, \rho, \varphi)$, where $U$is a$T^{n}$-invariant open set of_$X,$

$\rho$is

an

automorphism

of$T^{n}$, and

$\varphi$ is a $\rho-$equivariant diffeomorphism $homU$ to

some

$T^{n}$-invariant open subset in $\mathbb{C}^{n}$. The action of$T^{n}$ on $X$ is said to be locally standard if every point in $X$ lies in

some

standard coordinate neighborhood. See _{$[4, 2]$} for

more

details.

(Atypical example of locally standard torus actions is anonsingulartoricvariety.)

The atlaswhich consists of standard coordinate neighborhoods is weakly standard.

Therefore, a locally standard $T^{n}$-action induces the local $T^{n}$-action on _$X$

.

Notice that not all local torus actions

are

induced by locally standard torus actions. For any $C^{r}$ local $T^{n}$-action $\mathcal{T}$

on

a $2n$-dimensional manifold $X$,

we

take

a weakly standard atlas $\{(U_{\alpha}^{X}, \varphi_{\alpha}^{X})\}_{\alpha\in A}$ belonging to $\mathcal{T}$

.

It is easy to

see

that the

automorphisms $\rho_{\alpha\beta}$ of

$T^{n}$ in the property (2) of Definition 2.1 form

a

\v{C}ech

one-cocycle $\{\rho_{\alpha\beta}\}$

on

$\{U_{\alpha}^{B}\}_{\alpha\in A}$ with values in _$Aut(T^{n})$

.

Then, the cohomology class of $\{\rho_{\beta\alpha}\}$ in the first

\v{C}ech

cohomology set _{$H^{1}(B_{X_{1}}\cdot Aut(T^{n}))$} is the obstruction for the local $T^{n}$-action to be induced by a locally standard $T^{n}$-action.

Proposition 2.7. A $C^{r}$ local$T^{n}$-action on$X$ is induced by

some

$C^{r}$ locally

stan-dard$T^{n}$-action

if

and only

if

$\{\rho_{\alpha\beta}\}$ andthe triunal

\v{C}ech

one-cocycle are

of

the

same

equivalence class in$H^{1}(B_{X}; Aut(T^{n}))$, where the trivial

\v{C}ech

one-cocycle isthe

one

whose values on all open set are equal to the identity map

_of

$T^{n}$

.

For theproof,

see

[13].

Example 2.8. We

can

construct

an

example of local torus actionswhich does not

come

from any locally standardtorus fibrations in the following way. For

a

small positive number $0<\epsilon\ll 1$, let $\overline{X}$

be the quotient space ofthe space

$\{(z,w)\in \mathbb{C}^{2}\cross \mathbb{C}:0<|z_{1}|^{2}<1+\epsilon, |w|^{2}+|z_{2}|^{2}=1\}$

bythe $S^{1}$-action defined by

$u\cdot(z, w):=((z_{1},u^{-1}z_{2}),$$u^{-1}w$

).

$T^{2}$ actson $\overline{X}$ by

$u\cdot[z,w]$ $:=[u\cdot z,w]$

.

The map $\mu_{\overline{X}}:\overline{X}arrow \mathbb{R}^{2}$ defined by$\mu_{\overline{X}}([z, w])$ $:=(|z_{1}|^{2}, |z_{2}|^{2})$ is invariant under the

$T^{2}$-action and induces the identification of the orbit space of the $T^{2}$-action with

$(0,1+\epsilon)\cross[0,1]$

.

We define that two elements $\overline{x}_{1}$ and $\overline{x}_{2}$ in

$\overline{X}$

are

equivalent,

or

$\overline{x}_{1}\sim x^{\overline{x}_{2}}$ if for

a

representative $(z,w)$ of$\overline{x}_{1},$ $((\overline{z_{1}}/|z_{1}|\sqrt{|z_{1}|^{2}+1},\overline{z_{2}}),\overline{w})$ is

a

representative of$\overline{x}_{2}$

.

It does not depend

on

the choice of representatives of $\overline{x}_{1}$ and it is well-defined.

We denote the quotient space $\overline{X}/\sim x$ of the equivalence relation by $X$. By the

construction,

we

can

show that $X$ is endowed with

a

local $T^{n}$-action. The orbit

space $B_{X}$ is the cylinder defined by

$B_{X}$ $:=(0,1+\epsilon)\cross[0,1]/\sim B$,

where $\xi\sim B\eta$ if andonly if$\eta_{1}=\xi_{1}+1$ and $\eta_{2}=\xi_{2}$, and $\mu_{\overline{X}}$ inducesthe orbit map

$\mu x:Xarrow B_{X}$

.

Example 2.9 (Locally toric Lagrangian fibrations [7]). Let $\omega_{C^{n}}$ $:= \sqrt{2\pi-}^{1}\sum_{k\simeq 1}^{n}$ $dz_{k}\wedge d\overline{z}_{k}$ bethe standard symplectic structure

on

$\mathbb{C}^{n}$

.

Thestandardrepresentation

of $T^{\mathfrak{n}}$ preserves

$\omega_{C^{n}}$ and the map $\mu_{C^{n}}$: $\mathbb{C}^{n}arrow \mathbb{R}^{n}$ defined by (2.1) is

a

moment map of the standard representation of$T^{n}$

.

Notice that the image of$\mu_{C^{n}}$ is the $r\triangleright$

dimensional standardpositive

cone

$\mathbb{R}_{+}^{n}$

.

Let (X,$\omega$) be

a

$2n$-dlmensional symplectic

manifold and $B$

an

n-dimensional manifold with

corners.

A map $\mu:(X,\omega)arrow B$

is called

a

locally toric Lagrangian

_fibration

ifthere exists

a

system $\{(U_{\alpha},\varphi_{\alpha}^{B})\}$ of

coordinate neighborhoods of $B$ into $\mathbb{R}_{+}^{n}$, and for each $\alpha$ there exists a

symplecto-morphism $\varphi_{\alpha}^{X}$: $(\mu^{-1}(U_{\alpha}),\omega)arrow(\mu_{C^{n}}^{1}(\varphi_{\alpha}^{B}(U_{\alpha})),\omega_{C^{n}})$ such that

$\mu_{C^{n}}\circ\varphi_{\alpha}^{X}=\varphi_{\alpha}^{B}\circ\mu$

.

We show in [13] that fora$10$cally toric Lagrangian fibration$\mu:(x_{(v})arrow B$on

an

n-dimensional base $B$ and

an

above atlas $\{(U_{\alpha}, \varphi_{\alpha}^{B}, \varphi_{\alpha}^{X})\}$,

on

each nonempty overlap

$\varphi_{\alpha}^{X}o(\varphi_{\beta}^{X})^{-1}$ is $\rho$-equivariant. (Precisely, $\rho_{\alpha\beta}$ is

a

map from $U_{\alpha}\cap U_{\beta}arrow Aut(T^{n})$

.

Since $Aut(T^{n})$ is discrete, $\rho_{\alpha\beta}$ is locally constant.) In particular, $X$ is endowed with a smooth local $T^{n}$-action. In Section 6, we will describe the necessary and

sufficient condition that

a

manifold with

a

localtorus action becomes alocally toric Lagrangian fibration.

3. CHARACTERISTIC PAIRS AND CANONICAL MODELS

In this section,

we

introduce the characteristic pair for

a

local torus action,

and construct the canonical model from the characteristic pair. Both of them

play important roles of the topological classification of local torus actions. Inthis

section, all manifolds, maps, and local $T^{n}$-actions are assumed to be of class $C^{0}$

unless otherwise stated.

3.1. Characteristic $pai\iota \bm{s}$

.

Let $B$ be an$n$-dimensional topological manifold with

comers.

We

aesume

that $\partial B\neq\emptyset$. By the definition of amanifold with corners, $B$

is equipped with anatural stratification. We denote by $S^{(k)}B$ _the k-dimensional

stratum of$B$, namely, $S^{(k)}B$ consists of those pointswhich have exactly $k$

nonz

_$ero$

componentsinalocalcoordinate. $\bm{t}$particular, thetop-dimensionalstratum$S^{(n)}B$ is equal to the interior $B\backslash \partial B$ of $B$

.

Let $\Lambda$ _{$:=\{t\in \mathfrak{t}:\exp t=1\}$} be the lattioe ofintegral elements in the Lie algebra

$t$of $T^{n}$

.

Since the differential ofany automorphism of$T^{n}$ at the unit element

pre-serves

$\Lambda$, by associating any automorphism of $T^{n}$ with its differential at the unit

element, there is thenatural homomorphism from $Aut(T^{n})$ to $GL(\Lambda)$

.

It is an

iso-morphism. Infact, itfollows from the surjectivity of the exponential map of$T^{n}$ and

the equatIon $\varphi o\exp=exp\circ d\varphi$for any automorphism $\varphi\in Aut(T^{n})$. In the rest of

this paper,

we

ldentify $Aut(T^{n})$ with$GL(\Lambda)$ bythis isomorphism. Let $\pi_{P}$: $Parrow B$

be aprincipal $Aut(T^{n})$-bundle

on

$B$ and $\pi_{\Lambda}$: $\Lambda_{P}arrow B$ the associated

$\Lambda$-bundle of

$P$ by the above isomorphism $Aut(T^{n})\cong GL(\Lambda)$

.

Suppose that $\pi_{\mathcal{L}}$: $\mathcal{L}arrow S^{(n-1)}B$

is arank one sub-bundle of the restriction $\pi_{\Lambda}|_{S(n-1)B}$: $\Lambda_{P}|_{S(n-1)B}arrow S^{(n-1)}B$ of

$\pi_{\Lambda}$: $\Lambda_{P}arrow B$ to $S^{(n-1)}B$. For each

$k$ and any point $b\in S^{(k)}B$, let $U$ be an open

neighborhood of$b$ in $B$

on

which there exists alocal trivialization $\varphi^{\Lambda}$: $\pi_{\Lambda}^{-1}(U)arrow$

$U\cross\Lambda$ of$\Lambda_{P}$

.

By shrinking $U$ if necessary,

we

ct

assume

that the intersection

$U\cap s^{(n-1)B}$ _of $U$ with $S^{(n-1)}B$ has exactly $n-k$ connected components, say,

$(U\cap S^{(n-1)}B)_{1},$

$\ldots,$ $(U\cap S^{(n-1)}B)_{n-k}$

.

Since

$\Lambda$ isdiscrete, foreach $(U\cap S^{(n-1)}B)_{a}$

there exists arank

one

sub-lattice $L_{a}\subset\Lambda$ such that $\varphi^{E}$ sends the preImage

$\pi_{\mathcal{L}}^{-1}((U\cap S^{(n-1)}B)_{a})$ of$(U\cap S^{(n-1)}B)_{a}$ by$\pi_{\mathcal{L}}$ fiber-wiselyto $(U\cap S^{(n-1)}B)_{a}\cross L_{a}$

.

Deflnition 3.1. $\pi_{\mathcal{L}}$:

$\mathcal{L}arrow S^{(n-1)}B$ is said to be unimodularif for each $k$ and any

point $b\in S^{(k)}B$, thesub-lattice$L_{1}+\cdots+L_{n-k}$ generated by$L_{1},$

$\ldots,$ $L_{n-k}$ isarank

$n-k$directsummandofA. (In [4] suchasub-latticeis called

an

$(n-k)$-dimensional unimodular subspaceofA.)

Notice that rank

one

sub-lattices $L_{1},$

$\ldots,$ $L_{n-k}$ depend

on

the choice of

a

neigh-borhood $U$ and

a

localtrivialization$\varphi^{E}$

.

But Definition 3.1 does notdependon the

choice of them because theconditionfor

a

sub-latticeto be unimodular is invariant by

an

automorphism ofA.

Definition 3.2. Let $\pi_{\mathcal{L}}$: $\mathcal{L}arrow S^{(n-1)}B$ be

a

unimodular rank

one

sub-bundle of

$\pi_{\Lambda}|_{S(n-1)B}$ : $\Lambda_{P}|_{S(n-1)B}arrow S^{(n-1)}B$

.

Thenthepair $(P, \mathcal{L})$ of the principal$Aut(T^{n})-$

bundle $\pi_{P}$: $Parrow B$ and $\pi_{\mathcal{L}}$:

$\mathcal{L}arrow S^{(n-1)}B$ is called a characteristic pair and

$\pi_{\mathcal{L}}$:

$\mathcal{L}arrow s^{(n-1)B}$ is called a characteristic bundle.

Let (X,$\mathcal{T}$) be

a

2

$n$-dimensional manifold equipped with

a

local $T^{n}$-action. We

showthat there isacharacteristic pair associated with (X, 7‘). Let $\{(U_{\alpha}^{X}, \varphi_{\alpha}^{X})\}_{\alpha\in A}\in$

$B_{X}$ which satisfies the properties in Remark 2.4 and also determines a

\v{C}ech

one-cocycle $\{\rho_{\alpha\beta}\}$

on

$\{U_{\alpha}^{B}\}_{\alpha\in A}$ with coefficients in _$Aut(T^{n})$

.

It defines the principal

$Aut(T^{n})$-bundle $\pi_{P_{X}}$ : $P_{X}arrow Bx$ on $B_{X}$ by setting (3.1) $P_{X}$ $:=(I_{\alpha}I^{U_{\alpha}^{B}}\cross Aut(T^{n})$

ノ

$/\sim P$

where $(b_{\alpha}, h_{\alpha})\in U_{\alpha}^{B}\cross Aut(T^{n})\sim P_{X}(b_{\beta}, h_{\beta})\in U_{\beta}^{B}\cross Aut(T^{n})$ if and only if $b_{\alpha}=b_{\beta}\in U_{\alpha\beta}^{B}$ and $h_{\alpha}=\rho_{\alpha\beta}\circ h_{\beta}$

.

The bundle projection $\pi_{P_{X}}$ is defined by

the obvious way. For each $\alpha$, every point in $\pi_{P_{X}}^{-1}(U_{\alpha}^{B})$ has aunique representative

which lies in $U_{\alpha}^{B}\cross Aut(T^{n})$. By associating apoint in $\pi_{P_{X}}^{-1}(U_{\alpha}^{B})$ with the unique

representative, we define the local trivialization of $P_{X}$ on $U_{\alpha}^{B}$ which is denoted

by $\varphi_{\alpha}^{P}$ : $\pi_{\overline{p}_{X}^{1}}(U_{\alpha}^{B})arrow U_{\alpha}^{B}\cross Aut(T^{n})$

.

Let $\pi_{\Lambda_{X}}$ : $\Lambda xarrow B_{X}$ be the $\Lambda$-bundle

as-sociated with $P_{X}$ by the natural identification $Aut(T^{n})\cong GL(\Lambda)$

.

The property

(2) in Definition 2.1 $deter\min\oplus a$ _unlque unimodular sub-bundle of the restriction

$\pi_{\Lambda x}|_{S(n-1)B_{X}}$ : $\Lambda x|_{S(n-1)B_{X}}arrow S^{(n-1)}B_{X}$ of$\pi_{\Lambda_{X}}$

:

$\Lambda_{X}arrow B_{X}$ to the codimension

one

stratum $S^{(n-1)}B_{X}$ in the following way. For each coordinate neighborhood

$(U_{\alpha}^{B},\varphi_{\alpha}^{B})$ of_$Bx$ with$U_{\alpha}^{B}\cap S^{(n-1)}B_{X}\neq\emptyset$, thepreimage$\mu_{\mathbb{C}^{n}}^{1}(\varphi_{\alpha}^{B}(U_{\alpha}^{B}\cap S^{(n-1)}B_{X}))$

is equipped with the $T^{n}$-action which is the $restr\ddagger ction$ of the standard represen-tation of $T^{n}$

.

For simplicity,

we assume

that the intersection $U_{\alpha}^{B}\cap S^{(n-1)}B_{X}$

is connected. (Otherwise,

we

may consider component-wise.) Then, all points of

$\mu_{\mathbb{C}^{n}}^{1}(\varphi_{\alpha}^{B}(U_{\alpha}^{B}\cap S^{(n-1)}B_{X}))$ has the

common

one-dimensional stabilizer withraepect

to the $T^{n}$-action. We denote it by $S_{\alpha}^{1}$ and also denote the rank

one

sub-lattice of$\Lambda$

spanned bytheintegral element whichgenerates $S_{\alpha}^{1}$ by $\mathcal{L}_{\alpha}$. Supposethat $(U_{\alpha}^{B}, \varphi_{\alpha}^{B})$

and $(U_{\beta}^{B}, \varphi_{\beta}^{B})$ are coordinate neighborhoods satisfying the above conditions and

the intersection $U_{\alpha\beta}^{B}\cap S^{(n-1)}B_{X}$ is nonempty. Since the overlap map $\varphi_{\alpha\beta}^{X}$ is

a

$\rho_{\alpha\beta}$-equivariant homeomorphism,

we can

show that $\rho_{\alpha\beta}$ sends $S_{\beta}^{1}$ isomorphically to $S_{\alpha}^{1}$

.

Under the identification of

$\rho_{\alpha\beta}$ with the automorphism of$\Lambda indu\infty d$ by $\rho_{\alpha\beta}$, $\rho_{\alpha\beta}$ also

sends

$\mathcal{L}_{\beta}$ isomorphically to $\mathcal{L}_{\alpha}$

.

By the construction of $\pi_{\Lambda x}$

:

$\Lambda_{X}arrow B_{X}$

,

$\varphi_{\alpha}^{P}$ induces alocal trivialization $\varphi_{\alpha}^{\Lambda}$:

$\pi_{\Lambda_{X}}^{-1}(U_{\alpha}^{B})arrow U_{\alpha}^{B}\cross\Lambda$ of $\pi_{\Lambda_{X}}$ : $\Lambda_{X}arrow B_{X}$

on

each $U_{\alpha}^{B}$ such that on

an

overlap $U_{\alpha\beta}^{B}$ the transition function with resp$ect$ to $\varphi_{\alpha}^{\Lambda}$ and $\varphi_{\beta}^{\Lambda}$ is

$\rho_{\alpha\beta}$

.

We take asubsystem

$\{(U_{\alpha_{l}}^{B}, \varphi_{\alpha:}^{B})\}_{i\in \mathcal{I}}$ of$\{(U_{\alpha}^{B}, \varphi_{\alpha}^{B})\}_{\alpha\in A}$ which

covers

$S^{(n-1)}B_{X}$ and define the rank

one

sub-bundle $\pi_{\mathcal{L}_{X}}$: $\mathcal{L}_{X}arrow S^{(n-1)}B_{X}$ of

$\pi_{\Lambda_{X}}|_{S(n-1)B\chi}$: $\Lambda_{X}|_{S\langle \mathfrak{n}-1)B_{X}}arrow S^{(n-1)}B_{X}$ by setting

(3.2) $\mathcal{L}_{X}$ 一

$(\coprod_{i}U_{\alpha}^{B}$. $\cap S^{(n-1)}B_{X}\cross \mathcal{L}_{\alpha}:)/\sim L$

where $(b_{i}, l_{i})\in U_{\alpha}^{B}:\cap S^{(n-1)}B_{X}\cross \mathcal{L}_{\alpha}:\sim L(b_{j}, l_{j})\in U_{\alpha_{j}}^{B}\cap S^{(n-1)}B_{X}\cross \mathcal{L}_{\alpha_{j}}$ if

and only if $b_{i}=b_{j}$ and $l_{i}=\rho_{\alpha:\alpha_{j}}(l_{j})$

.

By the construction, it is easy to

see

that $\pi_{\mathcal{L}x}$: $\mathcal{L}_{X}arrow S^{(n-1)}B_{X}$ is unimodular. As

a

summary,

we

have the following

proposition.

Proposition 3.3. Associated rvith

a

local$T^{\mathfrak{n}}$-action$\mathcal{T}$

on

$X$, there exists a

charac-teristicpair$(P_{X}, \mathcal{L}_{X})_{f}$ where$P_{X}$ and$\mathcal{L}_{X}$

are

defined

by (3.1) and(3.2), respectively.

Notice that the characteristic bundle is a generalization of the characteristic

function of

a

quasi-toric manifold,

or

a

torus manifold.

Example 3.4. For a $2n$-dimensional manifold $X$ equipped with

a

locally

stan-dard $T^{n}$-action,

$\pi_{P_{X}}$ : $P_{X}arrow B_{X}$ is the trivial principal $Aut(T^{n})$-bundle $Px=$

$B_{X}\cross Aut(T^{n})$

.

Let $(S^{(n-1)}B_{X})_{a}(a=1, \ldots, k)$ be the connected component

$(S^{(n-1)}B_{X})_{a}$ by $\mu_{X},$ $T^{n}$-action

on

it has the unique one-dimensional stabilizer

which we denote by $S_{a}^{1}$. Let $\mathcal{L}_{a}$ be the rank

one

sub-lattice in $\Lambda$ corresponding to $S_{a}^{1}$

.

Then, $\mathcal{L}_{X}$ is the disjoint union $LI_{a}(S^{(n-1)}B_{X})_{a}\cross \mathcal{L}_{a}$.

Example 3.5. In the

case

ofExample 2.8, the characteristic pair is constructed

as

follows. We identify A with $\mathbb{Z}^{2}$

and also identify $Aut(T^{2})$ with $GL_{2}(\mathbb{Z})$. Then $P_{X}$

can

be written by

$P_{X}=((0,1+\epsilon)\cross[0,1]\cross GL_{2}(\mathbb{Z}))/\sim P$

where $(\xi, A)\sim P(\eta, B)$ if and only if$\eta\sim B\xi$ and $B=-A$

.

The bundle projection

is defined by the obviousway. $\Lambda_{X}$ iswritten by the similar way, namely,

$\Lambda_{X}=((0,1+\epsilon)\cross[0,1]\cross \mathbb{Z}^{2})/\sim\Lambda$,

where $(\xi, m)\sim P(\eta, n)$ if and only if$\eta\sim B\xi$ and $n=-m$. With this notation, $\mathcal{L}_{X}$

is written by

$\mathcal{L}_{X}=((0,1+\epsilon)\cross\{0,1\}\cross\{0\}\oplus \mathbb{Z})/\sim\Lambda$ .

For $i=1,2$, let $B_{i}$ be

an

n-dimensional topological manifold with

corners

and

$(P_{i}, \mathcal{L}_{i})$

a

pair of

a

principal $Aut(T^{n})$-bundle

$\pi_{P_{1}}$ : $P_{i}arrow B_{i}$ and

a

unimodular rank

one

sub-bundle $\pi_{\mathcal{L}_{*}}$. :

$\mathcal{L}_{i}arrow S^{(n-1)}B_{i}$ of the restriction of the associated $\Lambda-$

bundle $\pi_{\Lambda_{:}}$ : Ap: $arrow B_{i}$ of $P_{i}$ bythe natural identification $Aut(T^{n})\cong GL(\Lambda)$ to the

codimension

one

stratum $S^{(n-1)}B_{i}$ of$B_{i}$

.

Definition3.6. An isomorphism$f_{P}$: $(P_{1},\mathcal{L}_{1})arrow(P_{2}, \mathcal{L}_{2})$from $(P_{1}, \mathcal{L}_{1})$ to $(P_{2}, \mathcal{L}_{2})$

is

a

bundleisomorphism$f_{P}$: $P_{1}arrow P_{2}$which

covers a

stratification preserving

home-omorphism $f_{B}$: $B_{1}arrow B_{2}$ suchthat the latticebundleisomorphism $f_{\Lambda}$ : $\Lambda_{P_{1}}arrow\Lambda_{P_{2}}$

induced by$f_{P}$ sends$\mathcal{L}_{1}$ isomorphicallyto $\mathcal{L}_{2}$

.

$(P_{1}, \mathcal{L}_{1})$ and $(P_{2}, \mathcal{L}_{2})$ are isomo_$7P$hic

if there exists an isomorphism between th$em$

.

The isomorphism class of the characteristic pair $(Px, \mathcal{L}x)$ is

an

invariant of a

local $T^{n}$-action

on

$X$

.

Lemma 3.7. For $i=1,2$, let $(X_{i}, \mathcal{T}_{i})$ be a $2n$-dimensional

manifold

$X_{i}$ Utth

a

local $T^{n}$-action $\mathcal{T}_{i}$.

If

there is

a

$C^{0}$ isomo_$rp$hism $f_{X}$: $(X_{1}, \mathcal{T}_{1})arrow(X_{2}, \mathcal{T}_{2})$, then

$f_{X}$ induces the isomorphism $f_{P_{X}}$ : $(P_{\lambda_{1}’}, \mathcal{L}_{X_{1}})arrow(P_{X_{2}}, \mathcal{L}_{X_{2}})$ between characteristic

pairs associated unth$X_{1}$ and $X_{2}$.

Proof.

Let $\{(U_{\beta}^{X_{1}}, \varphi_{\beta}^{X_{1}})\}_{\beta\in \mathcal{B}}\in \mathcal{T}_{1}$ and $\{(U_{\alpha}^{X_{2}}, \varphi_{\alpha}^{X_{2}})\}_{\alpha\in A}\in \mathcal{T}_{2}$ be maximal weakly standard atlases of$X_{1}$ and $X_{2}$, and $\{(U_{\beta}^{B_{1}}, \varphi_{\beta}^{B_{1}})\}_{\beta\in B}$and $\{(U_{\alpha}^{B_{2}}, \varphi_{\alpha}^{B_{2}})\}_{\alpha\in A}$ atlases of $B_{X_{1}}$ and $B_{X_{2}}$ induced by $\{(U_{\beta}^{X_{1}}, \varphi_{\beta}^{X_{1}})\}_{\beta\in B}$ and $\{(U_{\alpha}^{X_{2}}, \varphi_{\alpha}^{X_{2}})\}_{\alpha\in A}$, respectively. Suppose that $fx:(X_{1}, \mathcal{T}_{1})arrow(X_{2}, \mathcal{T}_{2})$ is a $C^{0}$ isomorphism and $f_{B}$ is the

home-omorphism from $B_{X_{1}}$ to $B_{X_{2}}$ which is induced by $f_{X}$

.

By definition, on $e$ach

nonempty overlap $U_{\beta}^{B_{1}}\cap f_{B}^{-1}(U_{\alpha}^{B_{2}})$, there exists

an

automorphism $\rho_{\alpha\beta}^{f}$ of$T^{n}$ such

that $\varphi_{\alpha}^{X_{l}}of_{X}\circ(\varphi_{\beta}^{X_{1}})^{-1}$ is $\rho_{\alpha\beta}^{f}$-equivariant. It is easy to

see

that the following

equality holds

(3.3) $\rho_{\alpha_{0},\beta_{0}}^{f}\circ\rho_{\beta_{0}\beta_{1}}^{X_{1}}=\rho_{\alpha_{0}\alpha_{1}}^{X_{2}}0\rho_{\alpha_{1}\beta_{1}}^{f}$

on

a

nonempty intersection $U_{\beta_{0}\beta_{1}}^{B_{1}}\cap f_{B}^{-1}(U_{\alpha_{O}\alpha_{1}}^{B_{2}})$, where $\rho_{\beta_{0}\beta_{1}}^{X_{1}}$ and $\rho_{\alpha 0\alpha_{1}}^{X_{2}}$

are

au-tomorphisms of $T^{n}$ in (2) of Definition 2.1 with respect to $X_{1}$ and $X_{2}$,

respec-tively. We define the bundle isomorphism $(f_{P})_{\alpha\beta}$: $U_{\beta}^{B_{1}}\cap f_{B}^{-1}(U_{\alpha}^{B_{2}})\cross Aut(T^{n})arrow$

$f_{B}(U_{\beta}^{B})$ $\cap U_{\alpha}^{B_{2}}\cross Aut(T^{n})$ by

$(f_{P})_{\alpha\beta}(b, h)$ $:=(f_{B}(b),\rho_{\alpha\beta}^{f}oh)$

.

By (3.3), we

can

patchthem together to obtain thebundleisomorphism$f_{P}$: $P_{X_{1}}arrow$

3.2. Canonical models. In [4, Section 1.5], Davis-Januszkiewicz

constructed

the canonical model of a quasi-toric manifold from the based polytope and the

char-acteristicfunction. A similar construction

can

be done by using the characteristic pair inthe following way. Let$B$ be

an

n-dimensional $C^{0}$ manifoldwith

corners

and

$(P, \mathcal{L})$

a

characteristicpair

on

$B$. We denoteby$\pi_{T}$ : $T_{P}arrow B$ the$T^{n}$-bundle

associ-ated with $P$ bythenatural action of_$Aut(T^{n})$on $T^{n}$. First

we

shallexplain that for

any k-dimensional part $S^{(k)}B,$ $(P, \mathcal{L})$ determines

a

rank$n-k$ sub-torus bundle of

the restriction of $\pi\tau$ : $T_{P}arrow B$ to $S^{(k)}B$

.

Let $\{U_{\alpha}\}$ be

an

open covering of$B$ such

that

on

each $U_{\alpha}$ there exists a local trivialization $\varphi_{\alpha}^{P}$: $\pi_{P}^{-1}(U_{\alpha})arrow U_{\alpha}\cross Aut(T^{n})$

.

On each nonempty overlap $U_{\alpha\beta}$

we

denote by $\rho_{\alpha\beta}$ the transition functlon with respect to $\varphi_{\alpha}^{P}$ and $\varphi_{\beta}^{P}$, namely,

$\varphi_{\alpha}^{P}\circ(\varphi_{\beta}^{P})^{-1}(b, f)=(b, \rho_{\alpha\beta}f)$

for $(b, f)\in U_{\beta}\cross Aut(T^{n})$

.

Notice that $p_{\alpha\beta}$ is locally constant since $Aut(T^{n})$ is

discrete. $\varphi_{\alpha}^{P}$ induces the local trivializations ofthe aesociated bundles $T_{P}$ and $\Lambda_{P}$

which are denoted by $\varphi_{\alpha}^{T}$ : $\pi_{T}^{-1}(U_{\alpha})arrow U_{\alpha}\cross T^{n}$ and $\varphi_{\alpha}^{\Lambda}$ : $\pi_{\Lambda}^{-1}(U_{\alpha})arrow U_{\alpha}\cross\Lambda$, respectively. For $S^{(k)}B$

we

take $U_{\alpha}$ with $U_{\alpha}\cap S^{(k)}B\neq\emptyset$

.

By replacing $U_{\alpha}$ by a

sufficiently smalloneif necessary,

we

may

assume

that theintersection$U_{\alpha}\cap S^{(n-1)}B$

of $U_{\alpha}$ with the codimension one part $S^{(n-1)}B$ of $B$ has exactly $n-k$ connected components, say $(U_{\alpha}\cap S^{(n-1)}B)_{1},$

$\cdots,$ $(U_{\alpha}\cap S^{(n-1)}B)_{n-k}$. For $k=n$, this meao

that $U_{\alpha}$ is contained in $S^{(n)}B$

.

For $k<n$, there

are

$n-k$ rank

one

sub-lattices

$L_{1},$

$\ldots,$ $L_{n-k}$ of

$\Lambda$ such that for $a=1,$

$\ldots,$

$n-k\varphi_{\alpha}^{\Lambda}$ sends the restriction of

$\pi_{\mathcal{L}}$: $\mathcal{L}arrow S^{(n-1)}B$ to $(U_{\alpha}\cap S^{(n-1)}B)_{a}$ isomorphically to the trivial rank one

sub-bundle $(U_{\alpha}\cap S^{(n-1)}B)_{a}\cross L_{a}$ of $(U_{\alpha}\cap S^{(n-1)}B)_{a}\cross\Lambda$

.

Since $\mathcal{L}$ is unimodular, $L_{1}$,

. .

.,

$L_{n-k}$ generate the $(n-k)$-dimensional sub-torus of $T^{n}$ which is denoted by

$Z_{U_{\alpha}\cap S^{(k)}B}$

.

For$k=n$

, we

define$Z_{U_{\alpha}\cap S^{(n)}B}$ tobe

the.trivial

subgroupwhichconsists

of the unit element. Notice that when $(P, \mathcal{L}),$ $\{U_{\alpha}\}$

,

and $\varphi_{\alpha}^{P}$

are

the

ones

induced

by

some

local $T^{n}$-action $\mathcal{T}$

on

$X,$

$Z_{U_{\alpha}^{B}\cap S^{(k)}B_{X}}$ is the

common

$(n-k)$-dimeoional

stabilizer of$T^{n}$-action on $\mu_{\mathbb{C}^{n}}^{1}(U_{\alpha}^{B}\cap S^{(k)}B_{X})$

.

Suppose that another $U_{\beta}$ satisfies the above condition and $U_{\alpha\beta}\cap S^{(k)}B\neq\emptyset$

.

By the definition of $(P, \mathcal{L}),$ $\rho_{\alpha\beta}$ sends $Z_{U_{\beta}^{B}\cap S^{(k)}B_{X}}$ isomorphicallyto $Z_{U_{\alpha}^{B}\cap S^{(k)}B_{X}}$

.

Hence, in the same way as before, they

are

patched togetherto form arank $n-k$

sub-torus bundle, which is denoted by $\pi z_{s^{(k)_{B}}}$ : $Z_{S^{(k)}B}arrow S^{(k)}B$, of the restriction

of $\pi_{T}$: $T_{P}arrow B$ to $S^{(k)}B$.

Definition 3.8. For $t,$ $t’\in T_{P},$ $t$ and $t’$

are

equivalent or $t\sim_{can}t’$ if and only if

$\pi_{T}(t)=\pi_{T}(t’)$ and $t’t^{-1}\in\pi_{Z_{s^{(k)_{B}}}}^{-1}(\pi_{T}(t))$ when $\pi_{T}(t)$ lies in $S^{(k)}B$. Notice that a

fiber of$\pi\tau:T_{P}arrow B$ is equipped with the structure of

a

group since its structure

group is $Aut(T^{n})$

.

We denote by $x_{(P,\mathcal{L})}$ the quotient space of$T_{P}$ bythe equivalencerelation. The

bundle projection $\pi_{T}$: $T_{P}arrow B$ descends to the map $\mu_{X_{(P.\mathcal{L})}}$ : $x_{(P,\mathcal{L})}arrow B$

.

On

any $U_{\alpha}$, under the identification $\varphi_{\alpha}^{T}$: $\pi_{T}^{-1}(U_{\alpha})arrow U_{\alpha}\cross T^{n}$, the equivalence $rela_{r}$.

tion in Definition 3.8

can

be rewritten

as

follows. For $(b, t),$ $(b’, t’)\in U_{\alpha}xT^{n}$,

$(b, t)\sim_{\epsilon an}(b’, t’)$ if and only if$b=b’$ and $t’t^{-1}\in Z_{U_{\alpha}\cap S^{(k)}B}$ when $b$lies in $S^{(k)}B$. Then, $\varphi_{\alpha}^{T}$ induces the identification of $\mu_{x_{(P,\mathcal{L})}}^{-1}(U_{\alpha})$ with $(U_{\alpha}\cross T^{n})/\sim_{can}$

on

$U_{\alpha}$

.

Now we take $\{U_{\alpha}\}$ to be

an

atlas $\{(U_{\alpha}, \varphi_{\alpha}^{B})\}$ of $B$

as

a manifold with

corners.

Since $\mathcal{L}$ is unimodular and $B$ is

a

manifold with corners, by the

same

way

as

in Davis-Januszkiewicz [4, Section 1.5], or Masuda-Panov [8, Section 3.2],

we

can

show that $(U_{\alpha}\cross T^{n})/\sim_{can}$ is also homeomorphic to

a

$T^{n}$-invariant open

sub-set $\mu_{C^{n}}^{1}(\varphi_{\alpha}^{B}(U_{\alpha}))$ of$\mathbb{C}^{n}$. Hence, by taking thecomposition of these identifications,

thereis

a

homeomorphism $\varphi_{\alpha}^{X_{(P_{X},\mathcal{L}_{X})}}$

$\varphi_{\alpha}^{B}$: $U_{\alpha}arrow\varphi_{\alpha}^{B}(U_{\alpha})$

.

Notice

that

on

$U_{\alpha\beta}$ the overlap map

with these identifications

isinducedby id$U_{\alpha\beta}\cross p_{\alpha\beta}$: $U_{\alpha\beta}\cross T^{n}arrow U_{\alpha\beta}\cross T^{n}$

.

Hence, $x_{(P,\mathcal{L})}$ is a$2n$

-dimensional

topological manifold equipped with a $C^{0}$ local $T^{n}$-action whose orbit space is $B$ and whose orbit map is $\mu_{X_{(P,L)}}$

.

Definition 3.9. We call$X_{(P,\mathcal{L})}$ the canonical model of $(P, \mathcal{L})$. In particular, when $(P, \mathcal{L})$ isthe characteristic pair $(P_{X}, \mathcal{L}_{X})$ ofalocal$T^{n}$-action$\mathcal{T}$

on

a$2n$-dimensional

manifold $X$,

we

also call $x_{(P_{X},\mathcal{L}_{X})}$ the canonical model of(X,$\mathcal{T}$).

The following propositions describe the properties of the canonical model. For

proofs

see

[13].

Proposition3.10. Foranycharacteristicpair$(P, \mathcal{L}),$

$\mu x_{(P,\mathcal{L})}$! $x_{(P,\mathcal{L})}arrow B$ admits

a

continu

ous

section $s$

.

For any characteristic pair $(P, \mathcal{L})$, recall that

a

fiber of$T_{P}$ admits

a

structure

of

a group. By the construction, a fiber of$\mu x_{(P,\mathcal{L})}$: $X_{(P,\mathcal{L})}arrow B$ also admits

a

group

structure.

Proposition 3.11 ([13]). For a $2n$-dimensional

manifold

(X,$\mathcal{T}$) equipped utth a

local$T^{n}$-action,

we

denote the associated$T^{n}$-bundle_{$T_{P_{X}}$}

of

$P_{X}$ by $\pi\tau_{x}$ : $T_{X}arrow B_{X}$

for

simplicity. Then $T_{\lambda’}$ acts

fiber-unse

on

X. Similarly $x_{(P_{X},\mathcal{L}x)}$ also acts

fiber-wise

on

X. For any$b\in B_{X}$ the action

of

$\mu_{X_{(P.\mathcal{L})}}^{-1}(b)$

on

$\mu_{X}^{-1}(b)$ issimply tmnsitive.

Thefollowing lemma followdirectly from the construction ofa canonical model.

Lemma 3.12. For

$i=1,2$

, let $B_{i}$ be an n-dimensional topological

manifold

with

comers

and $(P_{i}, \mathcal{L}_{i})$ a characteristic pair

on

$B_{i}$

.

Then, any isomo$rp$hism

$f_{P}$: $(P_{1}, \mathcal{L}_{1})arrow(P_{2}, \mathcal{L}_{2})$ induces the $C^{0}$ isomorphism $f_{X_{(P,\mathcal{L})}}$ : $x_{(P_{1},\mathcal{L}_{1})}arrow x_{(P_{2},\mathcal{L}_{2})}$

between canonical models

of

$(P_{1},\mathcal{L}_{1})$ and $(P_{2}, \mathcal{L}_{2})$

.

Remark 3.13. If there is

an

isomorphism $fp:(P_{1}, \mathcal{L}_{1})arrow(P_{2}, \mathcal{L}_{2})$ between

char-acteristic pairs, then the induced $C^{0}$ isomorphism

$fx_{(P,\mathcal{L})}$: $x_{(P_{1},\mathcal{L}_{1})}arrow x_{(P_{2},\mathcal{L}_{2})}$

between canonical models is fiber-wise group isomorphism.

4. THE EULER CLASSES OF ORBIT MAPS

In this section, for a local torus action we define the Euler class of the orbit

map as an obstruction class for the orbit map to have a continuous section. In

this section we

assume

that manifolds, maps, and local $T^{n}$-actions are of class $C^{0}$

unless otherwisestated. Let (X,$\mathcal{T}$) be

a

$2n$-dimensional manifold equipped with a

local$T^{n}$-action. We investigatewhen$\mu x:Xarrow B_{X}$ hasasection. Weassumethat

the indexset $\mathcal{A}$ofthe weaklystandard atlas $\{(U_{\alpha}^{X}, \varphi_{\alpha}^{X})\}_{\alpha\in A}$ is countableordered.

By the construction of $x_{(P_{X},\mathcal{L}_{X})}$, there exists

a

$C^{0}$ isomorphism $h_{\alpha}$: $\mu_{X}^{-1}(U_{\alpha}^{B})arrow$

$\mu_{X_{(PL)}}^{-1}x,x(U_{\alpha}^{B})$ covering the identity

on

each

$U_{\alpha}^{B}$ such that $h_{\alpha}$ is equivariant with

respect to the fiber-wise action of $T_{X}$

or

$x_{(P_{X},\mathcal{L}_{X})}$

.

(For example

we

can

take

$(\varphi_{\alpha}^{x_{1P_{X},\mathcal{L}_{X)}}})^{-1}0\varphi_{\alpha}^{X}$

as

$h_{\alpha}.$) On each nonempty overlap $U_{\alpha\beta}^{B}$ theequation (4.1) $h_{\alpha}\circ h_{\beta}^{-1}(x)=\theta_{\alpha\beta}^{X}(b)x$

for$b\in U_{\alpha\beta}^{B}$ and$x\in\mu_{X_{(P_{X},\mathcal{L}_{X})}}^{-1}(b)$ determinesauniquelocalsection$\theta_{\alpha\beta}^{X}$ of

$\mu x_{(P_{X}.\mathcal{L}_{X})}$

on

$U_{\alpha\beta}^{B}$

.

Let$S_{(P_{X},L_{X})}$ denote the sheaf of germsofcontinuous sections of$\mu x_{(P_{X},\mathcal{L}\chi)}$

.

Then local sections $\theta_{\alpha\beta}^{X}$ form a

\v{C}ech

one-chain $\{\theta_{\alpha\beta}^{X}\}$

on

$\{U_{\alpha}^{B}\}$ with values in

$s_{(P_{X},\mathcal{L}_{X})}$

.

$Mor\infty ver$, by definition, we

can

show the following lemma.

Let $H^{1}(Bx;S_{(P_{X},\mathcal{L}x)})$ denote theflrst $\check{C}e$chcohomology group

of$Bx$ with

val-ues in $S_{(P_{X},\mathcal{L}x)}$

.

By the above lemma, $\{\theta_{\alpha\beta}^{X}\}$ defines the cohomology class in

$H^{1}(Bx;S_{(P_{X},\mathcal{L}_{X})})$

.

We denote it by $e_{orbit}(X)$

.

It is easy to

see

that $e_{orbit}(X)$ does

$notdependonthechoiceofh_{\alpha}sanddependsonlyontheloca1T^{n}- actiononX$

.

Definition 4.2. We call $e_{orb\iota’t}(X)$ the Euler class

of

$\mu_{X}$.

Notice that ifthe local$T^{n}$-action is induced byalocally

standard

$T^{n}$-action and

$\partial B_{X}=\emptyset$

,

then

$\mu_{X}$: $Xarrow B_{X}$ is a principal $T^{n}$-bundle. In this case, $e_{orbit}(X)$ is

nothing but the Euler class of the principal$T^{n}$-bundle.

Theorem 4.3. $\mu x:Xarrow B_{X}$ has

a

section

if

and only

if

$e_{orbit}(X)$ vanishes.

Example 4.4. For the $T^{n}$-action

on

a

complex n-dimensional, nonsingular toric

variety $X,$ $e_{orbit}(X)$ vanishes.

Example 4.5. For Example 2.8, $e_{orbit}(X)$ vanishes. In fact, we

can

defined the

section $s$ of$\mu x:Xarrow B_{X}$ by

$s([\xi_{1}, \xi_{2}])$ $:=[(\sqrt{\xi_{1}}, \sqrt{\xi_{2}}), \sqrt{1-\xi_{2}}]$

for $[\xi_{1},\xi_{2}]\in B_{X}$.

For $i=1,2$, let $B_{i}$ be

an

n-dimensional topological manifold with

corners

and $(P_{i}, \mathcal{L}_{i})$ a characteristic pair

on

$B_{i}$

.

Suppose that there exists

an

isomor-phism $f_{P}$: $(P_{1}, \mathcal{L}_{1})arrow(P_{2}, \mathcal{L}_{2})$. By Lemma 3.12, it induces the isomorphism

$f_{P}^{*}:$ $H^{1}(B_{2};\mathscr{L}_{(P_{2},\mathcal{L}_{2})})arrow H^{1}(B_{1}; S_{(P_{1},\mathcal{L}_{1})})$ between cohomology groups. In

partic-ular, by Lemma 3.7 and Lemma 3.12, a $C^{0}$ isomorphism _{$f_{X}$}: $(X_{1}, \mathcal{T}_{1})arrow(X_{2}, \mathcal{T}_{2})$

induces the isomorphism $f_{P_{X}}^{*}$: $H^{1}(B_{X_{2}} ; S_{(Px_{2},\mathcal{L}x_{2})})arrow H^{1}(B_{X_{1}} ; S_{(P_{X_{1}},\mathcal{L}x_{1})})$

.

Lemma4.6. For$i=1,2$, let$(X_{i}, \mathcal{T}_{i})$ be a$2n$-dimensional

manifold

equippedwith

a

local$T^{n}$-action.

If

there is a $C^{0}$ isomorphism _{$f_{X}$}: _{$X_{1}arrow X_{2z}$} then_{$f_{P_{X}}^{*}e_{or}u_{t}(X_{2})=$}

$e_{orbit}(X_{1})$.

5. THE TOPOLOGICAL CLASSIFICATION The following is the main $th\infty rem$ of [13].

Theorem 5.1 ([13]). For $i=1,2$, let $(X_{i}, \mathcal{T}_{i})$ be

a

$2n$-dimensional

manifold

$X_{i}$

with

a

local $T^{n}$-action

T.

$X_{1}$ and $X_{2}$

are

$C^{0}$ isomo$rp$hic

if

and only

if

there

exists

an

isomorphism $f_{P}$: $(P_{X_{1}}, \mathcal{L}_{X_{1}})arrow(P_{X_{2}}, \mathcal{L}_{X_{2}})$ between characteristic pairs

associated unth $X_{1}$ and $X_{2}$ such that $f_{P}^{*}e_{orbit}(X_{2})=e_{orbit}(X_{1})$

.

Moreover,

for

any

characteristic pair $(P, \mathcal{L})$ on an n-dimensionaltopological

manifold

$B$ utth

comers

and

_for

any element $e\in H^{1}(B;S_{(P,\mathcal{L})})$, there exists a $2n$-dimensional$C^{0}$

manifold

(X,$\mathcal{T}$) equipped vnth a $C^{0}$ local $T^{n}$-action whose characteristic pair and the Euler

class

_of

the orbit map

are

equal to $(P, \mathcal{L})$ and$e$, respectively.

The idea

_of

the proof. The only if part follows from Lemma 3.7 and Lemma 4.6.

The proof of the if part is similar to the proof of the classification of principal

bundles and the idea is

as

follows. Recall that bydefinition, $e_{orut}(X)$

measures

the difference between $X$ and $x_{(Px,\mathcal{L}x)}$

.

If there is an isomorphism $f_{P}$: $(Px_{\iota}, \mathcal{L}x_{1})arrow$

$(P_{X_{2}}, \mathcal{L}_{X_{2}})$ , then,byLemma3.12, $f_{P}$ induces the$C^{0}$ isomorphism from$x_{(Px_{1},\mathcal{L}x_{1})}$

to$x_{(P_{X_{2}},\mathcal{L}_{X_{2}})}$

.

Moreover, suppose that $f_{P}^{*}e_{orht}(X_{2})=e_{orbt}(X_{1})$

.

This

means

that

the difference between $X_{1}$ and $X_{(P_{X_{1}},\mathcal{L}_{X_{1}})}$ is

same as

the difference between

$X_{2}$

and$x_{(P_{X_{2}},\mathcal{L}_{X_{2}})}$ underthe identification$x_{(P_{i}c_{1},\mathcal{L}\chi_{1})}\underline{\simeq}x_{(Px_{2},\mathcal{L}x_{2})}$

.

Hence, $X_{1}$ is $C^{0}$

isomorphic to $X_{2}$

.

For

more

details,

see

[13].

We focus

on

the

case

of locally standard torus actions. We remark that if

a

manifold$X$ isequippedwith

a

locallystandard torusaction, then, $P_{X}$ is the trivial

bundle $P_{X}=B_{X}\cross Aut(T^{n})$

.

In this case,

we

can

obtain the followingcorollary. It

is

a

generalizatlon of the topological classification theorem for effective $T^{2}$-actions

on

four-dimensionalmanifolds without finitestabilizersby Orlik-Raymond [10] and

for quasi-toric manifolds by Davis-Januszkiewicz [4].

Corollary 5.2 ([13]). Locally standard torusactions are

_classified

bythe

character-isticbundle and the Euler class

_of

the orbit map up to equivarianthomeomorphisms.

6. LOCALLY TORIC LAGRANGIAN FIBRATIONS

Let (X,$\mathcal{T}$) be

a

2

$n$-dimensional smooth manifold equipped

with a

smooth local

$T^{n}$-action $\mathcal{T}$

.

In this section,

we

investigate the condition in order that $\mu x:Xarrow$

$B_{X}$ becomes a locally toric Lagrangian fibration.

Lemma 6.1. Suppose that there enists a symplectic structure $\omega$

on

$X$ and there

also exists

a

weakly standard atlas $\{(U_{\alpha}^{X}, \varphi_{\alpha}^{X})\}_{\alpha\in A}\in \mathcal{T}$

of

$X$ such that on each

$U_{\alpha f}^{X}\varphi_{\alpha}^{X}$ preserues symplectic forms, namely, $\omega=\varphi_{\alpha}^{X^{*}}w_{C^{n}}$

.

For each nonempty

overlap $U_{\alpha\beta}^{X}\neq\emptyset$

,

let $p_{\alpha\beta}\in Aut(T^{n})$ be the automorphism in (2)

of

Definition

2.1 utth respect to $\{(U_{\alpha}^{X}, \varphi_{\alpha}^{X})\}_{\alpha\in A}$

.

We identify

$\rho_{\alpha\beta}$ with an element

of

$GL_{n}(\mathbb{Z})$ by

the natural

identification

$Aut(T^{n})\cong GL_{n}(\mathbb{Z})$. Let $\{(U_{\alpha}^{B}, \varphi_{\alpha}^{B})\}_{\alpha\in A}$ be the atlas

of

$B_{X}$ induced by $\{(U_{\alpha}^{X}, \varphi_{\alpha}^{X})\}_{\alpha\in A}$

.

Then,

on

each nonempty overlap $U_{\alpha\beta}^{B}\neq\emptyset$, the overlap map $\varphi_{\alpha\beta}^{B}$: $\varphi_{\beta}^{B}(U_{\alpha\beta}^{B})arrow\varphi_{\alpha}^{B}(U_{\alpha\beta}^{B})$ is

of

the

form

(61) $\varphi_{\alpha\beta}^{B}(\xi)=\rho_{\alpha\beta}^{-T}(\xi)+c_{\alpha\beta}$

,

$for$

some

constant $c_{\alpha\beta}$, where

$\rho_{\alpha\beta}^{-T}$ is the transpose inverse

of

$\rho_{\alpha\beta}$

.

In particular,

$B_{X}$ becomes a smooth

manifold

ntth

comers.

Proof.

Let $\omega_{R^{n}xT^{n}}$ be the symplectic form

on

$\mathbb{R}^{n}\cross T^{n}$ which is defined by

$\omega_{\mathbb{R}^{n}xT^{n}}=\sum_{k=1}^{n}d\theta_{k}\wedge d\xi_{k}$,

where $(\xi_{1}, \ldots,\xi_{n})$ is the standard coordinates of $\mathbb{R}^{n}$ and $(\theta_{1}, \ldots, \theta_{n})$ is the angle

coordinates of $T^{n}$ with period 1, which

means

$(e^{2\pi\theta_{1}}, \ldots, e^{2\pi\theta_{n}})\in T^{n}$. First

we

focus on the interior of $B_{X}$

.

We

can

show that for each $\alpha$, there exists

a

sym-plectomorphism $\phi_{\alpha}$: $(\mu_{X}^{-1}(U_{\alpha}^{B}\backslash \partial B_{X}),\omega)arrow(\varphi_{\alpha}^{B}(U_{\alpha}^{B}\backslash \partial B_{X})\cross T^{n}, \omega_{R^{n}x}\tau\sim)$ such

that $pr_{1}o\phi_{\alpha}=\varphi_{\alpha}^{B}0\mu_{X}$ and

on an

overlap $U_{\alpha\beta}^{B}$, the overlap map $\phi_{\alpha\beta}$ $:=\phi_{\alpha}0\phi_{\beta}^{-1}$

is of the form $\phi_{\alpha\beta}(b, u)=(\varphi_{\alpha\beta}^{B}(b), \rho_{\alpha}\beta(u)u_{\alpha\beta}(b))$ for

some

map $u_{\alpha\beta}$: $U_{\alpha\beta}^{B}arrow T^{n}$,

where$pr_{1}$: $\varphi_{\alpha}^{B}(U_{\alpha}^{B}\backslash \partial B_{X})\cross T^{n}arrow\varphi_{\alpha}^{B}(U_{\alpha}^{B}\backslash \partial B_{X})$ is thenatural projectionto the first factor. For

more

details,

see

[13]. Then, by [11, Lemma 2.5],

on

each overlap

$U_{\alpha\beta}^{B}\backslash \partial B_{X}$the overlapmap$\varphi_{\alpha\beta}^{B}$ is of the form (6.1). Since $U_{\alpha\beta}^{B}\backslash \partial B_{X}$ isopen dense

in $U_{\alpha\beta}^{B},$ $\varphi_{\alpha\beta}^{B}$ should be of theform (6.1)

on

the whole $U_{\alpha\beta}^{B}$

.

$\square$

Deflnition 6.2. We call the atlas $\{(U_{\alpha}^{B}, \varphi_{\alpha}^{B})\}_{\alpha\in A}$ of$B_{X}$ in Lemma6.1 an integral

affine

structure compatible with $\{(U_{\alpha}^{X}, \varphi_{\alpha}^{X})\}_{\alpha\in A}$

.

Let $\{(U_{\alpha}^{X}, \varphi_{\alpha}^{X})\}_{\alpha\in A}\in \mathcal{T}$ be

a

weakly standard atlas of $X$

.

Suppose that the

induced atlas $\{(U_{\alpha}^{B}, \varphi_{\alpha}^{B})\}_{\alpha\in A}$of$B_{X}$ is

an

integralaffine structure compatiblewith

$\{(U_{\alpha}^{X}, \varphi_{\alpha}^{X})\}_{\alpha\in A}\in \mathcal{T}$

.

Lemma 6.3.

The

characteristic

bundle $\pi_{\mathcal{L}x}$ : $\mathcal{L}_{X}arrow S^{(n-1)}B_{X}$

admits

a

smooth

section which

generates

$\mathcal{L}_{X}$ fiber-wisely. In particular, $\pi_{\mathcal{L}_{X}}$ : $\mathcal{L}_{X}arrow S^{(n-1)}B_{X}$ is

Proof.

Let $(U_{\alpha}^{B}, \varphi_{\alpha}^{B})$ beacoordinate neighborhood of$B_{X}$ with $U_{\alpha}^{B}\cap S^{(n-1)}B_{X}\neq\emptyset$

.

We may assume that the intersection $U_{\alpha}^{B}\cap S^{(n-1)}B_{X}$ is connected. (Otherwise, we may consider component-wise.) As described in the construction of $\mathcal{L}_{X}$, the

local trivialization $\varphi_{\alpha}^{\Lambda}$ of$\Lambda_{X}$ sends $\pi_{\mathcal{L}_{X}}^{-1}(U_{\alpha}^{B}\cap S^{(n-1)}B_{X})$ isomorphicallyto $U_{\alpha}^{B}\cap$

$S^{(n-1)}Bx\cross \mathcal{L}_{\alpha}$, where $\mathcal{L}_{\alpha}$ is arank

one

sublattice ofA. Then there exists aunique

generator$u_{\alpha}$ of$\mathcal{L}_{\alpha}$ such that $\varphi_{\alpha}^{B}(U_{\alpha}^{B})$and$\varphi_{\alpha}^{B}(U_{\alpha}^{B}\cap S^{(n-1)}B_{X})$ lie in theupper half

space $\{\xi\in \mathbb{R}^{n} :\langle\xi, u_{\alpha}\rangle\geq 0\}$ and thehyperplane $\{\xi\in \mathbb{R}^{n} : \langle\xi, u_{\alpha}\rangle=0\}$

determined

by $u_{\alpha}$, respectively. Suppose that $(U_{\beta}^{B}, \varphi_{\beta}^{B})$ is another coordinate neighborhoods

satisfying the above conditions and the intersection $U_{\alpha\beta}^{B}\cap S^{(n-1)}B_{X}$ is nonempty.

Let $u_{\beta}$ be the corresponding generator of $\mathcal{L}_{\beta}$. Since the overlap map $\varphi_{\alpha\beta}^{B}$ is of

the form (6.1), $\varphi_{\alpha\beta}^{B}$ sends $\{\xi\in \mathbb{R}^{n} :\langle\xi, u_{\beta}\rangle\geq 0\}$ and $\{\xi\in \mathbb{R}^{n} : \langle\xi,u_{\beta}\rangle=0\}$

diffeomorphically to $\{\xi\in \mathbb{R}^{n} :\langle\xi, u_{\alpha}\rangle\geq 0\}$ and $\{\xi\in \mathbb{R}^{n} :\langle\xi, u_{\alpha}\rangle=0\}$, respectively.

In particular, this implies that $u_{\alpha}=\rho_{\alpha\beta}(u_{\beta})$

.

Thus $u_{\alpha}’ s$ form the required section

of$\mathcal{L}_{X}$

.

$\square$

By (6.1) the structure group ofthe cotangent bundle $T^{*}B_{X}$ is $GL_{n}(\mathbb{Z})$ and the

principal $Aut(T^{n})$-bundle $P_{X}$ is nothing but the frame bundle of$T^{\cdot}B_{X}$. Now we

have the following exact sequence of associated fiber bundles of$P_{X}$

$0arrow\Lambda_{X}rightarrow T^{*}B_{X}arrow T_{X}arrow 0$

.

As is well-known, $T$“$B_{X}$ is equipped with the standard symplectic structure, and

it is easy to

see

that the standard symplectic structure

on

$T^{*}B_{X}$ descends to the

symplectic structure

on

$T_{X}$, which is denoted by $\omega_{Tx}$,

so

that $\pi_{Tx}$: $(T_{X},\omega_{Tx})arrow$

$B_{X}$ is a nonsingular Lagrangian fibration. Moreover, we can show that following

lemma.

Lemma 6.4. The canonical model $x_{(P_{X},\mathcal{L}x)}$ becomes

a

smooth locally toric

La-grangian $fibmt_{\dot{b}}on$ on $B_{X}$

.

Roughly speaking, the proof is

as

follows. For each $U_{\alpha}^{B}$, the section of $\mathcal{L}_{X}$

defines

a

Hamiltonian action ofsome sub-torus of $T^{n}$

on

$\pi_{T_{X}}^{-1}(U_{\alpha}^{B})$. $X_{(Px,\mathcal{L}_{X})}$

can

be obtained by symplectic cutting techniqu$e$ with respect to these Hamiltonian

torus actions. For

more

details,

see

[13].

HYom Lemma 6.4, in particular, $h_{\alpha}$: $\mu_{X}^{-1}(U_{\alpha}^{B})arrow\mu_{X_{(P_{X}.\mathcal{L}_{X})}}^{-1}(U_{\alpha}^{B})$ in Section

4

can

be taken to be a $c\infty$ isomorphism which

covers

the identity on each $U_{\alpha}^{B}$ and $\theta_{\alpha\beta}^{X}$ defined by (4.1)

can

be also taken to be a

$c\infty$ local section of $T_{X}$ on $U_{\alpha\beta}^{B}$

.

Then the necessary and sufficient conditioninorderthat $\mu x:Xarrow Bx$ becomes a

locally toric Lagrangian fibration is given

as

follows.

Lemma 6.5. Let (X,$\mathcal{T}$) be

a

$2n$-dimensional smooth

manifold

equipped unth

a

smooth local$T^{n}$-action$\mathcal{T}$. There exists asymplectic structuoe$w$ on$X$ and there also

exists aweakly standard atlas $\{(U_{\alpha}^{X}, \varphi_{\alpha}^{X})\}_{\alpha\in A}\in \mathcal{T}$

of

$X$ such thaton each$U_{\alpha}^{X},$$w=$

$\varphi_{\alpha}^{X^{*}}\omega_{\mathbb{C}^{n}}$

if

andonly

if

the atlas$\{(U_{\alpha}^{B}, \varphi_{\alpha}^{B})\}_{\alpha\in A}$

of

$B_{X}$

induced

by$\{(U_{\alpha}^{X}, \varphi_{\alpha}^{X})\}_{\alpha\in A}$is

an

integral

_affine

structure compatible with$\{(U_{\alpha}^{X}, \varphi_{\alpha}^{X})\}_{\alpha\in A}$ and

on

each nonempty overlap $U_{\alpha\beta}^{B},$ $\theta_{\alpha\beta}^{X}$ is a Lagrangian section, namely, $(\theta_{\alpha\beta}^{X})^{*}\omega_{T_{X}}$ vanishes.

For nonsingularLagrangianfibrations, this resultisobtainedby Duistermaat [5]. See also [11], [9]. Recently,in [6] Gay-Symington showed the similarresultfor

near-symplectic

four-manifolds.

Finally

we

statethe classification theorem for locally toric Lagrangian fibrations. For

a

locallytoric Lagrangian fibration $\mu:(X,w)arrow B$,thelocalsections$\theta_{\alpha\beta}^{X}$ define

a

\v{C}ech

cohomology class $\lambda(X)\in H^{1}(B_{X} ; \ovalbox{\tt\small REJECT}_{T_{X}}^{Lag})$ of $B_{X}$ with values in the

sheaf

Theorem 6.6 ([1], [13]). Locally $tor\dot{b}C$ La9rangian

fibrations

are

classified

by

inte-gral

_affine

structures

on

the bases and $\lambda(X)$ up to fiber-preserving

symplectomor-phisms.

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